1 min read
| Status | Active — read 2026-10-20 (NOTE-984) in full; worth reading for QUESTION-217 and QUESTION-218 as the multilevel case of an operator-induced coarse-to-fine hierarchy. The levels V_j are spanned by the eigenvectors of a self-adjoint diffusion T with λ^t ≥ ε at dyadic t. They are exactly nested, the wavelet spaces W_j are spectral bands of about one octave, and a localized orthonormal basis of each level is built without computing eigenvectors. Read it knowing that the levels are spectral bands and not spatial blocks (the dyadic cubes only shape the basis); that nesting needs no spectral decay, which buys only compression; that the compressed algorithm as written thresholds T^{2^{j−1}}, not T^{2^j−1} as §5.4 and Fig. 1 say, and the paper's own Fig. 9 follows the former (THEORY-758); that the indexing of V_j is inconsistent between (4.8), §4.6 and Theorem 27; that only the uncompressed route (Theorems 27 and 29) has proofs, while the compressed scheme's error, localization and O(n log² n) cost are argued or observed; and that the non-self-adjoint extension is asserted, not carried out. |
| Version | 3 |
| Tags | mathematics, anthology-candidate |
| Cited as “paper” by | NOTE-984 — Coifman and Maggioni — Diffusion wavelets |
| Cited as “source” by | THEORY-758 — For a self-adjoint diffusion operator T with spectrum in [0, 1], the diffusion-wavelet levels are spectral subspaces spanned by the eigenvectors with λ^s ≥ ε for an increasing sequence of powers s, so they form an exactly nested chain of T-invariant subspaces whose differences are bands of the spectrum, about one octave each, not spatial blocks; Coifman and Maggioni's compressed algorithm, which squares the operator on each orthonormal basis, thresholds T^{2^{j−1}} at level j and not T^{2^j−1} as the paper states; and since each level is T-invariant, each is an exactly closed linear coarse-graining whose localized basis is a choice the level does not fix |
Coifman, R. R. & Maggioni, M. (2006). Diffusion wavelets. Applied and Computational Harmonic Analysis 21(1): 53–94. doi:10.1016/j.acha.2006.04.004.
Read in full in NOTE-984 (2026-10-20).
A2 recommended it at high priority: a hierarchy of scales induced by an operator’s persistent modes, to compare with the hierarchy learning or prediction favours. Bears on QUESTION-217. NOTE-675 already cites it without a LIT.